Search and drop-down menu plugins?

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I wanted to know if there any plugins that work like the search/dropdown on this site –

Each brand would have its own page, so I would want to be able to select and be redirected to a page on the site. I’m not looking for a plugin to sell anything, just want to see if there are any plugins that work similarly.

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Why isn’t my theme updating?! I’ve cleared my browser (Chrome) cache, my Wordpress cache, tried to use different themes…. Not sure what else to do here.

Can you know how to add icons before the names of the pages in the drop down list as shown in image 01 and how to add buttons to the end of the drop down menu as shown in image 02?